Problem: Simplify the following expression: $\sqrt{27} + \sqrt{75}$
Explanation: First, try to factor any perfect squares out of the radicals. $= \sqrt{27} + \sqrt{75}$ $= \sqrt{9 \cdot 3} + \sqrt{25 \cdot 3}$ Separate the radicals and simplify. $= \sqrt{9} \cdot \sqrt{3} + \sqrt{25} \cdot \sqrt{3}$ $= 3\sqrt{3} + 5\sqrt{3}$ Finally, simplify by combining the terms. $= ( 3 + 5 )\sqrt{3} = 8\sqrt{3}$